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Theorem undif5 4485
Description: An equality involving class union and class difference. (Contributed by Thierry Arnoux, 26-Jun-2024.)
Assertion
Ref Expression
undif5 ((𝐴𝐵) = ∅ → ((𝐴𝐵) ∖ 𝐵) = 𝐴)

Proof of Theorem undif5
StepHypRef Expression
1 difun2 4481 . 2 ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
2 disjdif2 4480 . 2 ((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)
31, 2eqtrid 2789 1 ((𝐴𝐵) = ∅ → ((𝐴𝐵) ∖ 𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cdif 3948  cun 3949  cin 3950  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-nul 4334
This theorem is referenced by:  cyclnumvtx  29820  fressupp  32697  elrgspnlem4  33249  sucdifsn2  38239
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